Double sequence, two sequences converge, but to different limits?

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Solution 1

A very simple example:

$$a_{m,n}=\begin{cases} 1,&\text{if }m\le n\\ 0,&\text{if }m>n\;. \end{cases}$$

If you write out the double sequence as an infinite array, it’s very easy to see what happens:

$$\begin{array}{ccc} 1&1&1&1&1&\ldots&\to&1\\ 0&1&1&1&1&\ldots&\to&1\\ 0&0&1&1&1&\ldots&\to&1\\ 0&0&0&1&1&\ldots&\to&1\\ 0&0&0&0&1&\ldots&\to&1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \downarrow&\downarrow&\downarrow&\downarrow&\downarrow&&&\vdots\\ 0&0&0&0&0&\ldots \end{array}$$

Solution 2

$$a_{n,m}=\frac{n}{n+m}$$

One limit is 0 another is 1.

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Updated on February 01, 2020

Comments

  • Admin
    Admin almost 4 years

    Let $(a_{m, n})_{m, n}$ be a "double sequence" of real numbers; that is, for every pair $(m, n) \in \mathbb{N} \times \mathbb{N}$, $a_{m, n}$ is a real number. Suppose that, for every $m$, $(a_{m, n})_n$ converges to a limit $x_m$, and that for every $n$ the sequence $(a_{m, n})_m$ converges to a limit $y_n$. What is an example in which $(x_m)_m$ and $(y_n)_n$ both converge, but converge to different limits, i.e.$$\lim_{m \to \infty} \lim_{n \to \infty} a_{m, n} \neq \lim_{n \to \infty} \lim_{m \to \infty} a_{m, n}?$$

    • BigbearZzz
      BigbearZzz almost 8 years
      I would advice that next time you should also provide your motivation for the question and your progress so far on the question. It'd would make people want to help you more.
    • Admin
      Admin almost 8 years
      @Giovanni That one asks about sufficient conditions for equality, and the answers offer no examples of the kind requested here.
    • Giovanni
      Giovanni almost 8 years
      @LiveForever: there is an example in the question..